Astrophysics Stellar Oscillation3 From Jcost

Stellar oscillation spacing

Stars vibrate. The Sun rings like a bell, and the frequencies of its pressure waves, called p-modes, carry information about its interior. The key observable is the large frequency separation, the average gap between successive overtone frequencies. For the Sun, that spacing is about 135 microhertz. Asteroseismologists measure it from light curves and use it to estimate stellar mass and age.

In Recognition Science, the framework derives physical constants from a single cost function J(x) = (x + 1/x)/2 - 1, which measures the price of recognizing a ratio x. The framework's library proves general facts about this cost: it is zero when the ratio is 1, nonnegative for positive inputs, and its canonical threshold phi - 3/2 is positive. Those three facts are theorems, checked by a machine, but they say nothing about stars on their own.

The module named Stellar_Oscillation3_FromJCost applies this cost to the ratio m/e, where m and e are meant to be two quantities in stellar oscillation, but the module never defines them in stellar terms. It proves the three general facts for that ratio, nothing more. The docstring records an idea: the solar spacing might equal phi^k times 10 microhertz, with phi^5 times 1.22 giving 135 microhertz, close to the measured value. That is a research note, not a result.

What the module actually establishes is a template. It shows that the cost function has the properties needed for a physical model, and it packages them into a certificate structure. But without a definition of m and e from stellar physics, the module does not derive the spacing. The idea remains a hypothesis, testable by matching the predicted spacing to more asteroseismic data.

noncomputable def cert : pMode3Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

This module does not derive the solar frequency spacing from stellar physics. The 135 microhertz agreement is a research note, not a proved theorem. The framework does not define m and e for stellar oscillations.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Astrophysics/Stellar_Oscillation3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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